EuCAP 2006 - European Conference on Antennas & Propagation

Session: Session 5A03A - Modular Software Tools for Antenna Analysis (15c)
Type: Oral Antenna
Date: Friday, November 10, 2006
Time: 08:30 - 12:20
Room: Risso 6
Chair: Vandenbosch & Mittra

Seq   Time   Title   Abs No
1   08:30   Goals And Structuring of the Antenna Software Initiative (ASI) within ACE
Vandenbosch, G.A.E.1; Gillard, R.2
1K.U.Leuven, BELGIUM;

ACE is a Network of Excellence (NoE) within the 6th framework of the European Union (EU). A NoE provides the means to establish integrated European research communities. ACE stands for 'Antenna Center of Excellence'.

One of the efforts being performed within ACE is the Antenna Software Initiative (ASI). The reason for this is the observation that although for a long time already there has been a strong coordination in fundamental antenna modeling problems in Europe, concerning the actual software itself, the European effort is still scattered.

In answer to this problem, within the ACE network in a first step an inventory was made of the software available among all partners of ACE, describing in detail the possibilities and limitations (capabilities to handle complex structures, environments, feeds, active components,...). This was the basis for all further work within the ASI.

Two high level activities have been performed after the inventory phase: 1. benchmarking of existing software codes, and 2. laying the foundations for a European software library. The pillars on which this library will be build are the following: 1. The Electromagnetic Data Interface (EDI). This activity consists of defining a standardized way of storing electromagnetic data (currents, fields, ...) in files (the so-called Data Dictionaries), and of developing the necessary software routines to do this automatically through a simple routine call. There will be a separate paper in this convened session treating this topic.

2. A better understanding of the problems encountered during the integration of actual software already available at this moment at a European level. Also concerning this topic, there will be a separate paper describing the integration activities finished within ACE 1.

3. Development of modular and hybrid techniques at a European level. The idea here is to work out procedures to develop these approaches in a "European" group rather than at the individual research group level, in this way taking into account different views, and taking into account the various constraints.

These activities will be explained in more detail in the paper. They have been performed by the partners within the framework of the ASI structure, a structure set up in a democratic way to cope with the most important bottle necks encountered in the procedure towards this European library. The structure involves several Working groups and Special Interest Groups. In the paper, also this structure will be explained in more detail.

2   08:50   Overview of the Software Integration Activities within ACE
Volski, V.1; Vandenbosch, G.A.E.2; Yang, J.3; Sipus, Z.3; Kildal, P.-S.3; Vipiana, F.4; Pirinoli, P.4; Vecchi, G.4; de Vita, P.5; de Vita, F.5; Baccarelli, P.6; Rius, J.M.7; Espinosa , H.7; Mattes , M.8; Valero , A.9; Persson, P.10; Freni, A.5
1Katholieke Universiteit Leuven (KUL), BELGIUM;
3Chalmers, SWEDEN;
6Sapienza, ITALY;

The ACE project initiated the start of several integration activities between European institutions involved in electromagnetic modeling of antennas with planar or conformal topologies. The goal of the integration activities was / is not to create a global software package that integrates the software of all partners, but the main idea was to initiate a long term process for antenna software integration activities within the European antenna community. During the first two years of ACE the integration activities were performed in several groups with a rather small number of partners in each group. The groups were formed by partners who wanted to integrate a specific approach developed by one partner into the software code of another partner. The topics addressed and the partners involved are listed below. There were integration projects:

- on planar and cylindrical antennas, KUL and Chalmers

- on the Multi-Resolution (MR) Approach, POLITO and UNIFI

- on the MR Approach, POLITO and KUL

- on Green's function exchange, KUL and UNIFI

- on an efficient computation of periodic Greens function in layered dielectric media, Sapienza and KUL

- on UPC block LU-solver, UPC, EPFL and UPV

- on conformal antenna software, Chalmers and KTH

One of the main problems that almost all groups encountered was how to transfer information between the different codes. This is due to the absence of standardized ways to describe electromagnetic quantities (currents, Green's functions, fields, and so on). In practice, the information is stored in different codes in a different way, and it is written to files using incompatible formats. This is of course directly related to the work performed on data dictionaries for different electromagnetic quantities. Although the groups worked rather independently, the interaction between groups was very noticeable and fruitful. The results of the on-going activities were reported to all partners during the ASI meetings three times a year. For instance the format for Greens functions in planar media proposed by UPC and EPFL was used as an example to construct the format for other Greens functions (2 dimensional and spectral).

At the first stage the integration was performed using the exchange of specific information between different codes. This step did not require a large modification of the existing codes and it allowed to produce very noticeable results in a very short period of time. For instance the efficiency of antenna software codes based on the MoM method with RWG basis functions can be greatly increased by implementing the MR approach developed at Polito. Two groups have successfully partially implemented these methods in their codes.

A very significant increase of the capabilities of a software code can be achieved using exchange of Greens functions. This type of exchange is not very widespread. Several groups have demonstrated that the exchange of Greens functions is a very simple way to implement dielectric layers or periodicity in the codes that use a mixed-potential formulation for electric fields. Moreover the integration activity between Chalmers and KUL has demonstrated that the exchange of Greens functions saves a lot of time in constructing the solution for very complex problems like the finiteness of conformal antennas.

In the paper a short but complete overview of all integration activities will be given.

3   09:10   The ACE Activity on Standardized File Formats for Electromagnetic Software
Frandsen, P.1; Martinaud, J.-P.2; Vandenbosh, G.3
1TICRA Engineering Consultants, DENMARK;
2THALES Airborne Systems, FRANCE;
3Department Of Electrical Engineering, Katholieke Universiteit Leuven, BELGIUM

At the outset of the ACE Antenna Software Initiative (ASI) a number of goals were formulated to foster and enhance cooperation and structuring among European players in the field of antennas and applied electromagnetic research. One of the key goals is development of procedures for integration and interoperability of software codes from Universities, Research Institutes and Industry. A very important aspect of software integration is the exchange of data among the software tools and the primary approach used for this purpose is the exchange of data in files. Hence data exchange via files remains a key issue and the development of a standardized file format has thus been a core activity in the ASI.

The software situation is characterized by many diverse parties that develop proprietary software. Most of this software uses unique file formats. It is often difficult and costly to transform data from one party's software to be used in another party's software. It is evident that a standardized file format, if accepted by the community at large, will lead to much easier interchange of data and foster increased cooperation between engineers and physicists working with antennas and electromagnetism, and a new file format is seen as a common platform for future cooperation. Evidently the ASI was the most adequate organization to address the problem of a new file format. The Electromagnetic Division of ESA/ESTEC has, however, also realized the data exchange problems. Further, as it was realized that the ACE resources would be too limited, ESTEC decided to initiate a project where data exchange standardization is a key part, the idea being that much more can be achieved by combining the available resources. Hence a Joint ACE ASI and ESTEC Activity on File Formats and Data Exchange was agreed upon.

Immediately it was realized that a simple traditional format based on a line-by-line descriptions would be insufficient. A more modern approach would be needed, and a group was formed for specification of requirements and identification of key tasks. The main steps of the development include the selection of data to be covered by the standard and the subsequent specification of the files contents and how data shall be interpreted. Also, the form and the organization of data in the files must be determined and tools for using the files must be developed.

The data sets to be included were proposed by the requirements group and discussed in the ACE ASI plenum. The six most important sets were selected for the first version of the standard. For each data set a group of partners from ACE ASI and other ACE activities were formed to make a so-called Data Dictionary containing an exact description of data in physical and mathematical terms. Meanwhile, the requirements group in close co-operation with ESTEC was defining the representation of data in files, and a language based on XLM has been specified. Finally, in order to ensure a consistent file access and handling, it was decided that an interface module denoted the Electromagnetic Data Interface (EDI) should be developed.

In the present presentation we shall discuss the background for the development of standard file formats and the organization and management required for such a rather complex activity, where many partners are involved. The Data Dictionaries will be outlined and one of these discussed in detail. Further, the XML format will be looked into and the EDI will be presented.

4   09:30   Large-Size Local-Domain Basis Function to Embody the Locality of High Frequency Diffraction in the Method of Moments
Ando, M.; Shijo, T.; Rodriguez, L.; Hirano, T.
Tokyo Institute of Techonology, JAPAN

The method of moments (MoM) and the high frequency diffraction analyses, such as the Geometrical Theory of Diffraction (GTD), are the methods for low and high frequency problems respectively. In the MoM all elements are mutually coupled while for the case of the GTD the elements are coupled locally (locality principle). Both methods have been well developed but theyhave almost independent histories due to the inherently different concepts.

This presentation reviews the relation between the two methodologies but embodying the local property of diffraction in the MoM. The cancellation of the mutual coupling is reproduced in the reaction matrix of the MoM due to the rapid phase change in the high frequency [1]. We introduce novel local-domain basis functions in the MoM that have domains in the order of wavelength, which are much larger than the segment length of wavelength/ 20 to wavelength/ 10 in the usual MoM. In addition, the phase detour is geometrically defined by the path length from a source included in thebasis functions. Here, the discretization interval of the MoM is not large and is standard, i.e., thebasis functions overlap densely. The large-size basis functions with the phase detour guarantee the rapid decay of the mutual coupling, excepting the GO components which would survive in the high frequency. Then the off-diagonal elements in the reaction matrix evanesce rapidly. Finally, the Fresnel zone number, which corresponds to the path length difference geometrically defined by the
positions of two segments of interest on the scatterer and the source, is shown to have a remarkable positions of two segments of interest on the scatterer and the source, is shown to have a remarkable correlation with the magnitude of matrix elements; a truncation criterion to make the sparse band matrix is obtained in terms of this number.

The use of large-size basis functions was firstly proposed in the Impedance Matrix Localization
(IML) method [2] and the phase detour in the Integral Equation-Asymptotic Phase (IE-AP) method was introduced by [3]. Our methodology improves the accuracy as well as the fidelity of the
canceling mechanism introducing the phase detour from a source, differing from [1] and [2]. The Dirac delta is used as the MoM weighting functions to obtain the canceling effect. The concept of
the high frequency is used in the MoM but differing from the hybrid methods where the basis functions with unknown weighting coefficients are allocated only in parts of the scatterer.

The MoM, with the localization implemented would be computationally light in the higher frequency. Results for 2-D scattering surfaces demonstrate the feasibility as well as difficulties of the
concept. The number of the important elements reduces from O(N^2) to O(N) for these specific examples.
[1] T. Shijo, et al. ''Large-Size c, '' IEICE Trans. Electron., Vol.E88-C, No.12, pp.2208-2215, Dec.
[2] F.X. Canning, ''The impedance matrix localization (IML)c,'' IEEE Antennas Propag. Mag.,
vol.32, no.5, pp.18-30,Oct.1990
[3] K.R. Aberegg, et al. ''Application of the integral equation-asymptotic phase methodc,'' IEEE
Trans. Antennas Propag., vol.43, no.5, pp.534--537, May 1995.

5   09:50   Hybrid Methods for Large Antennas and Periodic Arrays in Isolation and on Platforms
Volakis, J.; Lee, J.; Burkholder, R.; Sertel, K.
The Ohio State Univ, UNITED STATES

Recent development in the hybridization of the finite element (FE) and boundary integral (BI) methods have allowed for the rigorous analysis of large finite arrays by taking advantage of the repeated elements. This approach has also been extended to the analysis of periodic and finite volume metamaterial structures. The combination of such rigorous methods with high frequency techniques or iterative approaches allows for the evaluation of the array performance on large platforms.

This presentation will address the main features of the domain decomposition concept. Specifically, recent developments have allowed for the analysis of non-periodic meshes at the boundaries of the periodic elements [1]. In addition, the same decomposition approach has been adapted for the analysis of entire ships and aircraft. Examples of these will be presented at the meeting. The surrounding structures can be modeled using different techniques (moment method or high frequency techniques.), and be integrated with the full wave methods to iteratively obtain full-scale solutions [2]. We will present such integrations of these methods with specific emphasis on using the iterative physical optics (IPO) to carry out the large structure modeling [3]. When a moment method is used to model the platform, we will discuss a preconditioning approach to allow for concurrent high density gridding within the FE-BI domain whereas the platform is modeled with larger elements.

[1] Marinos N. Vouvakis, Kezhong Zhao, Seung Mo Seo, and Jin-Fa Lee, "Electromagnetic Analysis of Unbounded Problems Using a Domain Decomposition Based FEM-BEM Hybrid" submitted to IEEE Trans. Antenna and Propagat.
[2] R. J. Burkholder, P. H. Pathak, K. Sertel, R. J. Marhefka, J. L. Volakis, and R.W. Kindt, "A hybrid framework for antenna/platform analysis," Applied Computational Electromagnetics Society Journal, 2006.
[3] P. H. Pathak and R. J. Burkholder, "High frequency scattering," Book Chapter in Scattering: Scattering and Inverse Scattering in Pure and Applied Science, pp. 245-276, edited by E.R. Pike and P. Sabatier, Academic Press, Ltd., London, 2002.

6   10:40   Combining the Characteristic Basis Function Method with Rooftops and Razor-Blade Testing Functions over NURBS Patches
Delgado, C.1; Catedra, F.1; Mittra, R.2
1Alcala University, SPAIN;
2Pennsylvania State Univ, UNITED STATES

Recent research by a number of workers have shown that the Characteristic Basis Function Method (CBFM) is an efficient iteration-free approach for solving large problems that does not suffer convergence difficulties, often encountered by iterative Method of Moments (MoM) algorithms. The CBFM begins by subdividing the original geometry into manageable-size blocks, and then generates higher level macro-basis functions associated with these blocks. The method has also been found to be useful for hybridizing with Physical optics (PO) for multiscale geometries that have both smooth surfaces and regions with fine features. The purpose of this paper is twofold. First, we present a new implementation of the CBFM, where the Characteristic Basis Functions (CBFs) are stored as rooftops-generated from Non-Uniform Rational B-Splines (NURBS) in the parametric (u,v) domain-and the testing procedure makes use of razor-blade functions for the rooftops. Second, we utilize fast numerical techniques to perform matrix-vector products to speed up the computation of far-region reaction terms, needed for the computation of the reduced matrix which is subsequently solved directly without resorting to iteration. Figure 1. CBFs are expanded over a generic curved surface (a) using rooftops. (b) Razor-blade functions are involved in the testing process. Figure 2. Numerical example, monostatic RCS with two simply-curved surfaces (the electrical size is around 2.7), f=200MHz, =180, =0 to 90, - polarization

7   11:00   Recent Developments in Parallelized Softwares Based on Iteration-Free Techniques for Solving Very Large Problems
Mittra, R.1; Monorchio, A.2
1EMC Lab, Penn State University, UNITED STATES;
2University of Pisa, ITALY

Despite spectacular increases in our capability to numerically model, simulate the performance, and design complex electromagnetic systems in recent years, we continue to be challenged by the need to solve even larger and more complex problems-for instance antennas mounted designing on satellites, aircrafts or ships-than we have been able to handle in the past, and to do so more efficiently in order to reduce the simulation time. There are many competing CEM approaches at our disposal, and we have recently witnessed great progress made toward the enhancement of these numerical techniques. Some examples of such advances are the Fast Multipole Method (FMM) for Method of Moments (MoM) problems, and hybrid techniques that combine the asymptotic methods with numerically rigorous codes. We note that the integration of these tools can be carried out in a modular fashion, if desired. The paper will begin by first providing a number of practical examples that illustrate the fact there still exists a great need for codes that can handle problems involving: (a) very large number of degrees of freedom (DOFs); multiscale geometries; complex and inhomogeneous media, e.g. metamaterials. Next, it will present two approaches-one for the frequency domain and the other for the time domain-both of which depart from the conventional approaches to solving large problem, namely the use of iterative techniques. The two methods do share several common attributes: (i) they are highly parallelizable and, hence, suited for efficient solution of large problems; (ii) their modular nature builds on an existing MoM and FDTD codes; (iii) they are able to handle multiscale problems without running into convergence difficulties. The paper will provide some examples to illustrate the application of the CBMOM and SPFDTD software modules that have been developed for solving radiation, scattering and compatibility problems involving antennas. The presentation will conclude by predicting that the trend in CEM, which is expected to significantly impact the computing landscape in the near future points to increase the use of highly parallel computers, as well as, of network computing. It will when argue that we must follow this trend in order to be successful in our enterprise for modeling large antenna problems.

8   11:20   Overview of the Current State of Antenna Modelling and Development of Modular Software in Ukraine and the Former Soviet Union
Nosich, A.1; Kirilenko, A.A.2; Rud, L.A.2; Tkachenko, V.I.2
2Institute of Radio-Physics and Electronics NASU, UKRAINE

Three main parts of the state science in the USSR were: (1) fundamental science at the R&D institutes of the USSR Academy of Sciences, (2) defence and applied science at R&D institutes of the USSR Ministries, and (3) university science of the all kinds at the classical and technical state universities. Here, the Ukrainian science was standing out as it inherited some 25% of the total USSR staff and funding [1]; other non-Russian republics had only occasional laboratories. Antenna engineering and modelling had mostly followed this pattern, with main efforts concentrated in Moscow however still remarkable groups working in Kharkiv, Kiev, Dnipropetrovsk, and Lviv, and still others in Tbilisi and Minsk. After the dissolution of the USSR the funding for R&D had dropped down in 100 times and more. Therefore, e.g., R&D institutes of the Academy of Sciences have turned into unofficial technoparks where selected laboratories try to cope with life by doing essentially applied science and engineering.

Still within the last 5 years the Russian defence industry has obtained good funding, thanks to the multi-billion contracts from China, India, and Iran. Characteristically, the industries involved try to make all needed R&D themselves; as a result, for antenna modelling they use exclusively Western commercial software. In contrast, in Ukraine there are no industries fed with fat weapon contracts from the Middle or Far East. In parallel, both in Russia and Ukraine more elaborated antenna modelling methods are still developed at the same laboratories of the Academy of Sciences and at the universities as they were in 1991 although the scale of activities is much smaller now. Many of scientists traditionally work with singular integral equations and mode-matching techniques rather than with differential-equation formulations; normally they develop home-made software. It should be admitted, however, that only the software [2] has a modular structure. The main ideas behind this software are as follows: (i) decompose any waveguide circuit into a sequence of regular segments of simple and multi-connected waveguides, (ii) find their full-wave modal bases, (iii) calculate S-matrices of corresponding plane junctions, and (iv) assemble the S-matrix of the whole circuit. Longitudinal segmentation is a part of pre-processing, which also reveals possible symmetries, to be taken into account in the search of the modal bases. Besides, the connectivities of the segment cross-sections are determined, to enable further a calculation of all independent TEM modes. Finally, the cross-sections are decomposed to rectangular sub-domains and the domains of the overlapping of each two of the segment waveguides are automatically determined. All this is necessary as the input data for the computation of the S-matrices of plane junctions. To compute junctions with partial overlapping, virtual waveguides of zero length have to be introduced. Eigenvalue problems for determining the critical frequencies and modal fields in each of segmented waveguides are reduced to determinant equations.

The S-matrices of plane junctions are computed also by using the mode matching. Obtaining a full-wave S-matrix of a plane junction needs the calculation of the coupling integrals between all modes of connecting waveguides. Final step is automatic assembling of the partial S-matrices of all junctions into the S-matrix of the whole circuit. This corresponds to successive reduction of two neighbouring junctions or building more complicated combinations. Here, we represent the junctions and units of the circuit as a graph and perform reduction of this graph generated in the pre-processing of the circuit geometry. Various optimization strategies taking into account the geometry peculiarities and the available set of variable geometrical parameters are used in the course of graph reduction. As examples, two problem-oriented CADs with graphic user interfaces and vector analysis, synthesis, and optimization tools realized as modular software, are reported.

One of them deals with a full-wave model of a circular corrugated horn that provides the specified return loss, polarization purity, and radiation pattern symmetry over the two bands with a wide separation between them. The proposed synthesis procedure allows one to consider the horns with minimum number of slots in the throat and flare sections. Another is a highly efficient full-wave electromagnetic model and CAD of a septum polarizer with a square or circular output waveguide.

[1.] Nosich A.I., et al., IEEE Microwave Magazine, vol. 2, 82-90, 2002.
[2.] Kirilenko A.A., et al., in Hirshfield J. (Ed.), NATO ARW Quasioptical Control of Intense Microwave Transmissions, Springer, Berlin, 55-64, 2005.

9   11:40   Dual Wideband Radiating Element for Mobile Handsets for a European Antenna Software Benchmarking in ACE network
Ribero, J.-M.1; Staraj, R.1; Dubard, J.-L.1; Lethuc, P.1; Gillard, R.2; Paul, D. L.3; Herranz, J.I.4; Marrone, M.5


The Antenna Center of Excellence (ACE) Network was created on January 1st 2004 within the European Community 6th framework program in Information Society Technologies. The software activity is one of the most important activities in ACE. Its objective is to establish a list of existing software and estimate their performances by using test examples. Selected groups of software will also be combined and made available with documentation and support. To do so, three different work-packages corresponding to three distinct tasks have been planned:

- The inventory action first gives a general overview of the European antenna software domain.

- The benchmarking action proposes a set of structures to assess antenna software.

- The integration action studies possible combinations of software tools.

At present, for most antenna types, no common European criterion exists to estimate an antenna software tool based on its performances. One possibility to clarify this situation is benchmarking. Of course, performances will depend on the kind of investigated structure. In this paper, a dual wideband radiating element for mobile handsets proposed for the benchmark activity in ACE is presented. This action consists of a comparison between numerical techniques and experiment for this small, totally metallic, probe-fed antenna. The different institutions involved have simulated this structure with their own codes and/or commercial codes (LEAT, IDS, IETR, UOB, UPV).

Structure Description

The structure proposed for benchmarking is illustrated in the different figures below. The antenna is made up of two stacked quarter-wavelength elements short-circuited along a same plane. The patches are realized with rectangular 0.3 mm copper sheets on air substrate, to provide the largest bandwidth as possible for a total height of 11.6 mm. The originality of the structure also comes from the feed connected to the upper patch instead of the lower one (coaxial probe diameter: 1.2 mm).

Each resonator includes a slot possessing a special layout and its width is 0.3mm. This structure also corresponds to a realistic configuration of miniature antenna dedicated to mobile phone. The dimensions of the ground plane are close to the dimensions required in modern mobile phone handsets for the PCB supporting all electronic components and antenna. It is also a multiband structure, optimised for GSM900 (lower bandwidth), DCS/PCS and UMTS bands (upper bandwidth).

Results and Conclusion :

The proposed structure is interesting for benchmarking purposes especially due to the thin slots in the metallic parts. The fact that these slots are close to the vertical metallic parts of the antenna seems to create particular modelling difficulties as shown in the figures below. In fact, although all software agree for the upper band VSWR response, no software (in house or commercial) currently proves capable to model the lower band VSWR response therefore pointing out the need for further software development to tackle this type of antennas.

10   12:00   Integration of UPC AntennaLab Block-LU Direct Solver in MoM Codes
Rius, Juan M.1; Heldring, A.1; Espinosa, H.1; Mattes, M.2; Valero, A.3; Parron, J.4
1Universitat Politecnica de Catalunya, SPAIN;
2Ecole Polytechnique Fdrale de Lausanne (LEMA-EPFL), SWITZERLAND;
3Universitat Politecnica de Valencia, SPAIN;
4Universitat Autonoma de Barcelona, SPAIN

Software integration between Antennas Center of Excelence (ACE) partners is one of the main activities of the Antenna Software Initiative (ASI) work package. This paper will present the results of integrating the block-LU solver for very large systems of equations developed at UPC AntennaLab in the method of moments codes (MoM) of EPFL and UPV partners. It must be remarked that a non-ACE group (UAB) has also integrated the block-LU solver in his MoM code.

In recent years, a wide range of methods have been developed for accelerating the iterative solution of the discretized electromagnetic integral equations [1]. However, the linear systems arising from very large electromagnetic scattering and radiation problems are often poorly conditioned, especially when they are based on the EFIE, as is the case for problems involving open surfaces. This causes iterative methods to converge very slowly or not at all. Therefore it is crucial to use an efficient preconditioner or, in the last case, a direct (non-iterative) solution method.

LU decomposition is routinely used for either the direct solution of the linear system or to obtain an incomplete LU (ILU) decomposition of the pre-conditioner in iterative methods. However, the standard in-place LU decomposition algorithm needs to allocate the whole matrix in computer memory, which is not possible when the problem to be solved has a large number of unknowns.

To overcome this problem, UPC AntennaLab developed a block-LU algorithm [2] that performs a multilevel division of the matrix in blocks and sequentially applies the inverse-by-partitioning formula to each level. Unlike the conventional LU, the new approach allows swapping matrix blocks between core memory and hard disk. This procedure opens the door to solve large problems with moderate size computers and can be easily integrated into any MoM code.

[1] W. C. Chew, J-M. Jin, C.-C. Lu, E. Michielssen, and J. M. Song, "Fast Solution Methods in Electromagnetics," IEEE Transactions on Antennas and Propagation, Vol. 45, No. 3, pp. 533-543, March 1997.

[2] A. Heldring, J.M. Rius, L. Ligthart, "New block ILU preconditioner scheme for numerical analysis of very large electromagnetic problems", IEEE Trans. on Magnetics, Vol.38, No.2, pp. 337-340, March 2002.